Approximation of solution operators of elliptic partial differential equations by H- and H2-matrices
Approximation of solution operators of elliptic partial differential equations by H- and H2-matrices
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DOI：
10.1007/s00211-009-0278-7
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发表时间：
2010-04-01
影响因子：
2.1
通讯作者：
Boerm, Steffen
中科院分区：
文献类型：
--
作者：
Boerm, Steffen
我们研究了椭圆型偏微分方程有限元离散产生的刚度矩阵逆矩阵的计算问题。由于解算子是非局部的，逆矩阵通常是稠密的，因此通过标准技术表示它们将需要大量的存储。在积分方程领域，有效处理稠密矩阵的一种成功技术是使用数据稀疏表示，如流行的多极方法。在本文中，我们证明了这种方法可以推广到覆盖逆矩阵对应的偏微分方程切换到数据稀疏H-和H-2-矩阵。关键的结果是存在性证明的局部低秩近似的解决方案运营商和其离散对应，从而产生误差估计的H-和H-2-矩阵近似的整个矩阵。
We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse H- and H-2-matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for H- and H-2-matrix approximations of the entire matrices.